Congruence and Coincidence

Objective

SWBAT use a rigorous definition of congruence--aided by tracing paper, compass, etc.--to identify congruent figures. Students will understand the meaning of congruence in terms of superimposing figures with rigid motions.

Big Idea

Hands-on activities reinforce the connection between congruence and coincidence, helping students to go beyond "same shape, same size".

Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

Lesson Open

The warm-up prompt for this lesson asks students to define congruence. The warm-up follows our Team Warm-up routine. I choose students at random to write the team's answer on the board.

We review the team answers as a class. Most of my students will probably define congruent as "same shape, same size". I tell students that that definition is still true, but it is only part of the story. For example, what does it mean for objects to have the same shape? Students will probably agree that objects have the same shape and size if--and only if--one can be superimposed on the other so that all points coincide--line up. In geometry, that means we can describe a sequence of reflections, rotations, and translations that carries one object onto the other.

I display the agenda and learning targets for the lesson. Today we will focus on re-examining the concept of congruence in terms of the rigid motions.

Team Warm-up

Agenda and Learning Targets slide

Re-Examining Congruence slide show

Team Warm-up slide

Re-Defining Congruence

15 minutes

Old notions die hard, so I re-teach the concept of congruence explicitly with the aid of Guided Notes on Congruence and Symmetry. For more on how I use guided notes to teach, see this description of my thoughts on this instructional strategy: Guided Notes.

Displaying the slide, I tell students that I want them to think about how they can use transformations to tell whether objects are congruent. A range of tools is available for them to use, for example: compass, rulers. (I am looking for students to find their own methods of deciding whether objects are congruent (MP5). I make tracing paper available, but only after students have tried using other methods to identify congruent polygons in problem #3 of the set. I like tracing paper, because students can use it to represent rigid motions in a very concrete way. However, I want students to make other connections, as well. Among these, I want students to see that a compass performs transformations on a line segment. Also, I want students to make the connection between the congruence of segments and the equality of their lengths, and between the congruence of angles and the equality of their measures.)

I ask students to use Rally Coach, so that they can compare ideas and check each others' work. I hand out the problems on half-sheets of paper as I circulate around the classroom.

I am on the lookout for:

Do students see that a compass is a very efficient tool for comparing the length of segments (MP5)? I do not discourage students from using a scale to measure length, but I want to make sure that they are aware of the other tools in their toolbox.

In problem #2, do students see that angles may be congruent regardless of the apparent "length" of their sides? To highlight this point, I use an animated slide.

Are students using protractors correctly to measure angles? They should be extending the rays of the angles or sides of the polygons to measure accurately (MP6).

Matching Congruent Objects slide

Congruent Angles slide

Matching Congruent Objects

Rally Coach

Matching Congruent Parts

15 minutes

Displaying the slide , I tell students that now we will practice writing congruence statements and using congruence marks to show that objects are congruent. We will also practice matching up corresponding sides and angles of congruent polygons.

As with the last activity, I ask students to use Rally Coach, and I hand out the problems on half-sheets of paper as I circulate.

Be On the Lookout For: Do students see that they can use the congruence statement to identify pairs of corresponding vertices and sides (MP7)?

Corresponding vertices are given in the same order when naming congruent polygons to make it easier for the reader to follow. I will expect students to follow this convention, as well.

Matching Congruent Parts slide

Matching Congruent Parts handout

Lesson Close and Homework

5 minutes

Individual Size-Up

The lesson close follows our individual Size-Up routine. The prompt asks students to tell how they used transformations to determine whether figures are congruent during the lesson.

Homework

For homework, I assign problems #6-8 of Homework Set 1 for this unit. These problems provide additional practice in matching congruent segments, angles, and polygons and in identifying corresponding parts of congruent polygons.